In this work, we only consider the monochromatic PBH mass function. The extended mass function of PBHs can form extreme mass ratio inspiral, which can produce different signals. In detecting more PBHs events at high redshift, stochastic gravitational wave background from PBHs formation need to be deducted [1]}, [2]}, [3]}. Due to the high redshift of the gravitational wave, no electromagnetic counterpart can be observed along with the gravitational wave signal. We cannot directly observe the luminosity distance of gravitational wave. However, the interaction between Population III stars with PBHs could be a potential source for electromagnetic signals at \(z < 20\) , which helps us understand PBHs evolution at high redshift.
